HigmanSimsGroupHS

Details

By default, HigmanSimsGroupHS[] is represented as a permutation group acting on points {1,…,100}.

Background & Context

HigmanSimsGroupHS[] represents the Higman–Sims group , which is a group of order . It is one of the 26 sporadic simple groups of finite order. The default representation of HigmanSimsGroupHS is as a permutation group on the symbols having two generators.

The Higman–Sims group is the seventh smallest of the sporadic finite simple groups. It was introduced by mathematicians Donald G. Higman and Charles Sims in the late 1960s. HigmanSimsGroupHS is the simple index-2 subgroup of the automorphism group of the so-called Higman–Sims graph and contains a one-point stabilizer isomorphic to the Mathieu group . In addition to its numerous permutation representations, can be defined in terms of generators and relations as , with , , and . It can also be found as subgroups of various permutation groups such as ConwayGroupCo2 and ConwayGroupCo3, related to the so-called Leech lattice. Along with the other sporadic simple groups, played a foundational role in the monumental (and complete) classification of finite simple groups.

The usual group theoretic functions may be applied to HigmanSimsGroupHS[], including GroupOrder, GroupGenerators, GroupElements and so on. However, due its large order, a number of such group theoretic functions may return unevaluated when applied to it. A number of precomputed properties of the Higman–Sims group are available via FiniteGroupData["HigmanSims","prop"].